To find the equation of the new path that is perpendicular to the existing path and intersects at the point [tex]\((-2, -3)\)[/tex], we can follow these steps:
1. Identify the slope of the existing path:
The equation of the existing path is [tex]\( y = -2x - 7 \)[/tex].
This equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Here, the slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex].
2. Determine the slope of the new path:
Since the new path is perpendicular to the existing path, its slope will be the negative reciprocal of the slope of the existing path.
The negative reciprocal of [tex]\(-2\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
3. Use the point and the slope to find the equation of the new path:
We know that the new path passes through the point [tex]\((-2, -3)\)[/tex] and has a slope of [tex]\(\frac{1}{2}\)[/tex].
We can use the point-slope form of the equation of a line, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) = (-2, -3) \)[/tex] and [tex]\( m = \frac{1}{2} \)[/tex].
Substituting these values into the point-slope form, we get:
[tex]\[
y - (-3) = \frac{1}{2}(x - (-2))
\][/tex]
Simplifying this, we have:
[tex]\[
y + 3 = \frac{1}{2}(x + 2)
\][/tex]
Distribute the [tex]\(\frac{1}{2}\)[/tex] on the right-hand side:
[tex]\[
y + 3 = \frac{1}{2}x + 1
\][/tex]
Subtract 3 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{2}x + 1 - 3
\][/tex]
Simplify the right-hand side:
[tex]\[
y = \frac{1}{2}x - 2
\][/tex]
Therefore, the equation of the new path that is perpendicular to the existing path and intersects at the point [tex]\((-2, -3)\)[/tex] is:
[tex]\[
y = \frac{1}{2}x - 2
\][/tex]
